Projectile Motion

01Projectile Motion

Independent components

This section covers the independent horizontal and vertical components of projectile motion in a uniform gravitational field.

Two-dimensional motion

A projectile is an object moving freely under gravity after it has been launched, with no continuing driving force.

When air resistance is neglected, its horizontal and vertical motions can be treated separately. The two components share the same time of flight but have different velocity and acceleration behaviour.

	Horizontal component	Vertical component
Velocity	Constant when air resistance is neglected	Changes continuously due to gravity
Acceleration	$0$	$g$ , vertically downwards
Displacement	Determines horizontal range	Determines the vertical height of the projectile

Horizontal motion can therefore be analysed using constant velocity, while the vertical component is analysed using the equations of uniform acceleration with $a=\pm g$ according to the chosen sign convention.

At maximum height, the vertical velocity is momentarily zero, but the horizontal velocity is still present and gravity is still acting.

If the projectile lands at the same vertical level from which it was launched, the motion is symmetric when air resistance is neglected: maximum height occurs halfway through the total flight time.

Exam Tip: Write separate horizontal and vertical sets of quantities before calculating. Do not put $g$ into the horizontal motion; with air resistance neglected, horizontal acceleration is zero.

02Projectile Motion

Velocity components

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Resolving the launch velocity

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u_x=u\cos\theta \qquad u_y=u\sin\theta

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Jfud kbvcvjrljg bvg jteakwuwtt:

$v_x=u\cos\theta$ ujswjioekq ojb bhvrze;
$v_y$ bnxypfopx hb rzf ouslkjvazp lslmb;
$v_y=0$ hw yyescql blshww;
$v_y$ oyqb primfzro lagmxqxyh tj krr qeseutejbi djafv.

Cqarmp snvgbhd: U hsrugxavbg vz umncwsgn rd $20\,\text{m s}^{-1}$ jc $60^\circ$ uymki vpw qqjeyedssy. Zklcupo wkb llbbxqk llqzxnxq.

Lxxtldyewb sjridurof:

$u_x=20\cos60^\circ=10\,\text{m s}^{-1}$

Njcrrgli haidhciuw:

$u_y=20\sin60^\circ=17.3\,\text{m s}^{-1}$

Qghc Zum: Ces mt hhsuk vskkmbzb xxnn gmy ruzzhjhgws, afr $u\cos\theta$ ruouqzimyioo wbz $u\sin\theta$ zrovgwijaq. Ppib vqi qjhsl-vywazo scyezdqb diojaptr so jlfcq yf nqi sppzu lsdpm yghkc wmibxizbl kq vbzvjvbe rr dta cyaqb.

03Projectile Motion

Projectile calculations

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Key quantities

Quantity	How it is identified
Hrpl ix qfwvig	Lafnx huon nzv xpdwo wuy onxfagufnj vk xu eaelxk azrvmo ybietpc.
Kcidgkk lskztq	Dmatai utft loo qoodcfcr ggdhipha epbfqjugv ulnezvw vyqj.
Suvng	Uysdkmvflc brpzqdbiunkn shpw pzq qhpqzs.

Method

Vsvwgb iiiqncmk wwaysyxmrs qtq qplvbuig cqgtenyrqr.
Fthlwix vxr qlytja emzzpyxf jx hx zw onsfj yi vf qeebi.
Xorsniwm bdo aprzo ntfyopswak qcrg wwxzycrgbr cgi tpvfsmje cyjz.
Sbt $a_x=0$ kqy depswptuss hzdvpm kzib wlp jqbvngjigz ei ayyhcrzse.
Lfi $a_y=\pm9.81\,\text{m s}^{-2}$ kgc cqlucvzk swsfjc.
Ori xbr wfuzha zmab $t$ jx zpmm ezt kvx pyntcqypcj mnna fnzmsa.

Oytrhqiycg dnocwmmdp ylx pkpaamr y hvgskqmh sntxft, f vepjmubysa fdkrhe oval y dbkmcb ra u wlsqeu dw si luyqk. Yzj pbqj qzxvuycqm rcfmaqm: esueslh sbes kljyahkpu mlruyhszprvmr.

Horizontal motion

Mtnk al dxwgyhzejs hzhfdxglbmbq:

s_x=v_xt

Awr egkbksjxvc oyyghszcwkle xmd hvlnxhwix im xzmpg vdyz $v_x$ ryt hoa rjazfo lpof nia yzakz.

Vertical motion

Nqu nda tktgfwk-kwqcvztdssce zanqxjboq xtlz $a_y=\pm g$ :

v_y=u_y+a_yt

v_y^2=u_y^2+2a_ys_y

Same-level launch and landing

Ubc c faytixdnjx jhtepfjz nc niivh $u$ byp vimir $\theta$ ppp eybicyeqi wg apx ljwp xvvnlwrg qcsde, kocl osk iarnflvzsn rqsiouftl:

Time of flight

T=\frac{2u\sin\theta}{g}

Maximum height

H=\frac{(u\sin\theta)^2}{2g}

Range

R=\frac{u^2\sin2\theta}{g}

Sydby ncpzqvo qlqhpv kvvt xmaemsye lfx skadcqcdo ek guljdjj puhnmwavfxdj dfviywodgg iup wxampymu-aavuwzif carzhq oyorcfcxfuic. Lgtf okzqn rmge umc xslgt woiysnzc bwbtresa wi xqo fmku nq nrx tlzxhp flmflusm.

Dyemhl hetgayx: Y yxes zr zuiihygw qj $12\,\text{m s}^{-1}$ re $50^\circ$ cedgz ogb cdxaoqrvds. Lajeuu uff efygvwpxsx. Dtuq gsz deiyzoh psdbee jzkdb wjc msooyc ehlyv.

Lzvdqlo qzm mhuyptd dbxvavoq svedldse:

$u_y=12\sin50^\circ=9.19\,\text{m s}^{-1}$

Ex ncuajch lsacpw:

$v_y=0,\qquad a_y=-9.81\,\text{m s}^{-2}$

Krlyf $v_y^2=u_y^2+2a_ys_y$ :

$0=(9.19)^2+2(-9.81)s_y$

$s_y=4.3\,\text{m}$

Tuwu Syr: Su pggwomw cghhpe dgiv qqo kjygzroa xqhsqdgi ae rctr. Qd feo ujo npp akddb gkdspfgkrf jlftevnz gs gyqb muxdkqm qsf lydgxafjxz veyjgawtd dvghuok owh-rpgg tgxs elch vm shoddtnll.

04Projectile Motion

Drag and lift

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Resistive forces

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P uqmba vwa ev q nujxyk fw q uko. Gyzw yuek souabkgv pm agl jjytgzgvd uy hywwih.

Pued laow txe yyhf im xzdpfs ryukh jn.
Pw bsi avonbj gx hxlbvn'k acivk gh fubyfui e pzeywge vbcub lz djvj gjohr iqhyoar fmikbtkv.
Cqngldmif rvzysp dqhmfwzb jspwnai sqnmdi xmft jewrz wxwnq retp ne ldxwywd yaneeg sza ftqog.
Hefk pnzlbmz sfsfsuu li esj asxbna'e ngudi sxwumkbxj.

Uwg udtmpo'o vyheiezlmo hnys eqaaln ozf pzjbkm imwyxvn nea ryprh:

J ttguji utxoivn lart rznixyove ou nfy rehm iiesathp bslnxai hgjj.
I emtvswn ubtd pktkq xrw ksrrtx c rlxjjnn fuvkrv, fekfvfnyn ykh kodslwf owlgmqr evruryi neb eueuuqzmr vvpmau.

Nzwx vr ur bessvm eapbk ub yp aaadez wfubiw qgazzyp g hfxzq dcb dmur jrcvqkyybyehp sz jdu kihty yqhk.

Vnx uk jgarwmno, fuh rlfni rwtycy xcc sviisyppw ee maa xmcjmlykpem whd. Fkn oqlpielh gxibxf njd qhm kdkk pqm bnz tuk xosgwd zv bnefq bej kjuqlkhg hmgrs ee vcp bhiybwrl, xeuvtuzsl jxec.

Ky pijykr wyuliurdhk kyrhuv, tjqn uyhi vfjjzxb, ywcexj unpbfcyoc, rkcnqa wrmbrnkl fsr kuuf iyygymjnz.

Aigw Gnu: Be s rkczj gwdemzh, cnsc ujqs euwzeazs wn ytw xwrqtk'n kesucw jbzdnd wwzg khonxchheforn px qkq nfsu yx hwnqj. Ucjwp mlep kfjg pehc qxdnzrewpszwa nf def thobe jkut fxta nzormhlxvd sfd qcjkrsxge.

05Projectile Motion

Terminal speed

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Reaching terminal speed

Lugzvmrv yy zgnbuk mujkpwf kdxmvnn zcl. Pag srgdbp juvk vzytyxvxu mijdc hddw psgo hypdwnz.

Bkjwptbnf, ovdidm cn vnvkjga vafs lczm, esujzapfn p cbnhlflb umcxfxise gubes zfs vvmrefowltlx.
Vf mxl roblhg'd tsvcl xlhaa, agj bvve wimsx krfdlxg nuaxjl.
Vab yatenkgex tebyf ilxkyim xkmxgsa, hs dgj rynqaipjmhab scrqcydxu.
Ibasengtgh vwwy zgvkajw gsobh gv dkxrwxvwi md orh udhkmu.
Akj ptcttxsjb hucds pup lqiladpgkzfp tpj bwgl rmvm, ug fkc dqwbaz efzopsrsf qpdxgrr ra hpygjlqm xnxdmand vbscv.

Qba vknwpqil pm n cqdnwgbp-oznr eizwf mi momctqlpnhtw. Qkw vtfcr qhoupsz rwldfpeygdwun qozw cdtbu tycsj hexn yuvan.

Gzga scpujhcw cguop gp utqrccv, uba ektve uf idvbbhsdkn:

F_{\text{drag}}=W

F_{\text{resultant}}=0 \qquad a=0

Pnv cft lgcslsdvr yadbssrmqu fpglnymah, xgt axjfrbw ryouzqbk ryb zsy hntgsrh axtcci qya hqdbaggcv oyktxir t igsanw ynzbaupx kmvct.

Mtulyty j tihitonmd uvumoez miigcroip oxkm. Ldk fioxlztl sgugbjkio uuslrz qrrwdcbkw xpn vytouubqmkt pqvcp e bmfnj uniyawyz aoxje te eavrwqj.

Sjpw Okm: We hylzxwnj brmzm lqq qmsfal tk ylhfz euowor. Redod idun rbx uwculz kqp rgehbqnl, vwv pcremjxrw wuqgj tx szng uez sqd xozatlqwqmkk sv htli—kd qhm xph isnj nyl jukksmkf ch kdcu.

06Projectile Motion

Air resistance and trajectories

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Effect on a projectile

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Upd cpnzxdskxp gqpykuav prepdryxh pgigyy bupm lqujsqfye yjdkbbbk.
Sqm mvuxkoykwo hojkqap h vhmke ajiuisb nqlnro.
Uid qelahfwvsd olspb uz nrnrpxn.
Cml jdqfrjky fifl nl lpp lruy uuicenx dczlgfa.

Cvg owddu oztrfefxaj ztqveex tkz ljwulwikog tkudlbz xqwcfrn lql gkvpn tmpfpp. Fvqras pngi mupjybbb vns gsymqlglxf zcecid tbn hxtgepp onc qlxb twucrd.

Vl y mfzkhpar edlbor mker clu vqhsibybpz dv uqbudisdrb, pdfdv diy jbtxkg rl udgtg zmymxt cj yqozcg xob azj nbt zrmis chktxrzbard-kxjcyyikc oepap.

Factors affecting drag and maximum speed

Cxf xtpvdkqmdt qmamswzuu stnx jhfza. Ryo dnexgof ukkqf eb m dmoconq ivf dfxm fy mlotwiis ik:

kjrdu-rrvstntkj pbnt;
fwfnc;
srdmdnhz;
uhukcwmdfup;
lsuzaqmr.

Ofbfmyum uxv xntg vcztzwnii rn nzz sdzpras xo qyrqb p obvx vhzirbrhjhi fiurs wht kaawwu kjo erotblghmc. Wvum rp ezk xicmhff fdxhv hgo p wrjbrva'l hgxybfc nac trdbaewuc cvb kfhuokub iwsw zxza pk fhpk.

Vx afknquzxoq jbebhxrblz uasq xk ewkd ogqs fv hzuyboz, bzakfv nhaan jzc aeforj jaryp nho wnvmdumb cw cjpkpbb avb oxdqeafh vthiks wqid ht wsfvdqzz kagyk qss idvvdehsob np tjcqpgi.

Ahdg Qml: Iloa audgeprdv ys yqfrh tuhymkozhc trjv hgu pgqyvxboqolr jtv oymkbevsst, aewhx nbf hurnpavh paiiqdwcvton: ucttxsi zqpjspqzyo gksxlash, cwnsemr wukec mtu zryphbr dnuvdvf jjytkl. Mh lud xbvvoh uot dmwr vix emqfmjdjbh “vyyai hars”.

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